Mathematical Problems in Engineering

Volume 2017, Article ID 2968231, 18 pages

https://doi.org/10.1155/2017/2968231

## Prediction Model of Mechanical Extending Limits in Horizontal Drilling and Design Methods of Tubular Strings to Improve Limits

^{1}School of Aerospace Engineering, AML, Tsinghua University, Beijing 100084, China^{2}MOE Key Laboratory of Petroleum Engineering, China University of Petroleum, Beijing 102249, China

Correspondence should be addressed to Wenjun Huang; moc.621@6891nujnewgnauh

Received 15 February 2017; Revised 27 April 2017; Accepted 3 May 2017; Published 24 May 2017

Academic Editor: Mohammed Nouari

Copyright © 2017 Wenjun Huang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Mechanical extending limit in horizontal drilling means the maximum horizontal extending length of a horizontal well under certain ground and down-hole mechanical constraint conditions. Around this concept, the constrained optimization model of mechanical extending limits is built and simplified analytical results for pick-up and slack-off operations are deduced. The horizontal extending limits for kinds of tubular strings under different drilling parameters are calculated and drawn. To improve extending limits, an optimal design model of drill strings is built and applied to a case study. The results indicate that horizontal extending limits are underestimated a lot when the effects of friction force on critical helical buckling loads are neglected. Horizontal extending limits firstly increase and tend to stable values with vertical depths. Horizontal extending limits increase faster but finally become smaller with the increase of horizontal pushing forces for tubular strings of smaller modulus-weight ratio. Sliding slack-off is the main limit operation and high axial friction is the main constraint factor constraining horizontal extending limits. A sophisticated installation of multiple tubular strings can greatly inhibit helical buckling and increase horizontal extending limits. The optimal design model is called only once to obtain design results, which greatly increases the calculation efficiency.

#### 1. Introduction

Extending limit in horizontal drilling is a constrained optimization problem in which horizontal extension of the wellbore is the objective function, and kinds of ground and down-hole constraint conditions and operation conditions in horizontal drilling are considered. There are two main issues in extending limits including prediction and control. Prediction refers to “what are the extending limits,” and control means “how to increase the extending limits.” Until now, some researchers have built models by considering several factors or analyzed the extending limits with drilling data, and a systematic theoretical framework of extending limits in rotary drilling is forming.

Wu and Juvkam-Wold [1] studied the effect of helical buckling on the slack-off extension limit of tubing in horizontal wells and pointed out that helical buckling can limit wellbore extension a lot. Mason and Judzis [2] analyzed the extending limits in shallow, mid-deep, and deep wells with drilling data and the results show that friction force is main constraint factor for shallow wells, drill rig capacity and tubular strength are for mid-deep wells, and tubular strength is for deep wells. Wang and Guo [3] built the models of extending limits by considering the effects of formation property, hydraulic equipment capacity, drill fluid property, and so on. Gao et al. [4] proposed the concepts of extending limits of extended-reach wells. They pointed out that extending limits can be further divided into three categories according to the properties of constraint conditions, including mechanical extending limits, open-hole extending limits, and hydraulic extending limits, for the convenience of analysis. Mechanical extending limit means the maximum wellbore extension under the constraints of drilling mode, tubular strength, drill rig capacity, and so on, open-hole extending limit means the maximum wellbore extension within which either collapse or fracture of the wellbore does not happen, and hydraulic extending limit means the maximum wellbore extension in which cutting removing is available under certain hydraulic parameters. Yan et al. [5] built theoretical models to predict the extending limits by considering the effects of rock breaking threshold, tubular strength, friction factor, and so on. Li et al. [6, 7] studied the open-hole extending limits by considering the effects of well path, drill fluid property, wellbore collapse and fracture, and so on in kinds of drilling scenarios. However, the above studies mainly focus the value of extending limit for a certain well and neglect the distributions of extending limits for kinds of wells. In other words, the law of the envelope of the extending limits for certain drill rig and down-hole constraint conditions is not revealed.

Until now, methods to improve mechanical extending limits have been discussed in previous studies. Peng and Zhao [8] proposed that drill collars or heavy weight drill pipes should be installed in the vertical wellbore to provide enough weight on bit, and common and heavy weight drill pipes should be adopted on the horizontal or holding section to reduce the axial friction. Han [9] proposed that the design of tubular string components should include the effects of rock breaking threshold, tubular failure, and tubular buckling. According to Han’s studies, the objective functions of tubular string in sliding and rotary drilling modes should be, respectively, set to minimizing axial friction and minimizing tubular string weight. The tubular string components are constantly revised in iteration process. Allen et al. [10] studied the measures from the aspects of well path plan, drag reduction, and hydraulic parameters selection to improve extending limits with the case study of extended-reach wells in Wytch Farm. Xia et al. [11] proposed the design idea for tubular selection in extended-reach wells with trial and error method. The previous mechanical analysis of drill string behavior and design of tubular string components must be conducted several times to obtain an appropriate result in the previous methods, which means the design process is time-consuming and inefficient. Moreover, the previous design methods contain too much personal experience and lack of rigorous mathematical derivations.

In this paper, the mechanical extending limit model in horizontal drilling, in which kinds of ground and down-hole constraint conditions and operation conditions are included, is built. The proposed model is compared with other models in an example of four kinds of tubular strings. The law of distributions of mechanical extending limits is analyzed and the main factors constraining the wellbore extension are identified. A high-efficiency design method of drill strings components is built. At last, the extending limit model and design method are applied to a case study.

#### 2. Prediction Model of Mechanical Extending Limits

##### 2.1. Integral Mechanical Model

A typical horizontal well trajectory shown in Figure 1 includes three parts, namely, vertical, build-up, and horizontal wellbores. If a tubular string is in the pick-up process, the tubular strings in wellbores are in axial tensile state and the axis of tubular string is parallel to that of the wellbore. In the horizontal section, the tubular string lies on the bottom of wellbore due to gravity. In the build-up section, the tubular string touches the top or the bottom of the wellbore. In the vertical section, the tubular string stays on the wellbore axis and does not touch the inner surface of the wellbore. If the tubular string is in the slack-off operation, the mechanical behaviors of tubular strings become more complex. In the horizontal section, the tubular string is in axial compression state and the axial compressive force on the tubular string increases from the drill bit to the starting point of the horizontal section. If the axial force is smaller than the critical buckling load, the tubular string lies on the bottom of the wellbore. However, if the axial force exceeds the critical sinusoidal or helical buckling load, the tubular string enters into buckling state and the deflection curve of the tubular string is depicted by a sinusoidal curve or a helix. Therefore, buckling tends to occur on the starting part of the horizontal section shown in Figure 1. In the build-up section, the tubular string is usually in nonbuckling state for the buckling load is increased due to the effect of the curved wellbore configuration. Then, the tubular string tends to touch the bottom of the build-up wellbore. In the vertical wellbore, the axial compressive force increases with the increase of vertical depth. If the axial force is smaller than the critical helical buckling load, the tubular string keeps in straight line state. However, if the axial force exceeds the critical helical buckling loads, the tubular string becomes into helix state. Therefore, helical buckling tends to occur on the bottom part of the vertical wellbore.